Research

My research is in the representation theory of p-adic groups and automorphic forms, focusing on distinction problems, twisted Jacquet modules, and automorphic periods.

Preprints

[P3] H. Sharma, Duality, rigidity, and peeling for multisegments: on hypotheses of Mitra, Offen, and Sayag, arXiv:2609.21384 (2026). Submitted.

[P2] P. Dagar, H. Sharma, On representations of GL(n, D) admitting a generalized linear period, arXiv:2609.34307 (2026). Submitted.

[P1] S. Dhar, H. Sharma, Asai gamma factors and distinction in families, arXiv:2603.02699 (2026). Revision submitted to Journal of Number Theory.

Journal Articles

[5] H. Sharma, Symplectic model for Zelevinsky model of GL(n, D), Algebras and Representation Theory (2026). doi:10.1007/s10468-026-10421-5

[4] P. Dagar, H. Sharma, M. K. Verma, On representations of GL(n) distinguished by GL(1) × GL(n−1) over a quaternion division algebra, Journal of Algebra (2026). doi:10.1016/j.jalgebra.2026.07.003

[3] H. Sharma, M. K. Verma, Symplectic model for ladder and unitary representations, Journal of Pure and Applied Algebra (2026), Paper No. 108334. doi:10.1016/j.jpaa.2026.108334

[2] H. Sharma, On representations of GL(4, F) distinguished by GL(2, F) × GL(2, F), The Ramanujan Journal 67 (2025), Paper No. 91. doi:10.1007/s11139-025-01134-2

[1] H. Sharma, M. K. Verma, Symplectic period for a representation of GL(n, D), Journal of Pure and Applied Algebra 229 (2025), no. 6, Paper No. 107976. doi:10.1016/j.jpaa.2025.107976

Articles [1]–[3] are based on my Ph.D. thesis.

Current Research

(a) Symplectic periods of residual automorphic representations. Let F be a number field, D a division algebra over F, and D_𝔸 = D ⊗ 𝔸_F. I study which representations in the residual spectrum of GL(2n, D_𝔸) have a nonzero Sp(2n, D_𝔸)-period.

(b) Generalized linear periods for arbitrary n. I study which irreducible representations of GL(n, D) admit a nonzero (GL(1, D) × GL(n−1, D), χ_s)-equivariant linear form.

(c) Twisted Jacquet modules over division algebras. For principal series representations of GL(2n, D) induced from maximal parabolic subgroups, I study twisted Jacquet modules with respect to the (n, n) parabolic. I ask when they are nonzero, what their structure is, and what this gives for Shalika models.

(d) Twisted Gan–Gross–Prasad problem in low rank. I study existence and multiplicity of equivariant linear forms for the triple (GL(V), U(V), ω_{V,ψ,μ}) in low-rank cases.

Future interests. I want to learn geometric representation theory, especially p-adic Deligne–Lusztig theory. I would like to see whether its geometric and cohomological constructions give new approaches to distinction problems, for p-adic groups and for groups over division algebras.